Games and Puzzles
Mathematical games puzzles and problems
Circle Regions
Put n points on the circumference of a circle. Join each point to every other one with a straight line. This cuts the circle into a number of regions. What is the maximum number of regions you can generate from n points?
For example, with two points you just get two regions, with three points you get four regions, with four points you get eight regions.
Leapfrog
Adam and Brian decide to cycle from Adam’s house to Brian’s house 8 km away. Unfortunately Brian has left his bike at home! They decide to share Adam’s bike, each alternately riding and walking. They agree that the rider will get off the bike when he is a distance of h km ahead of the walker and will continue on foot himself. Once the walker reaches the bike he will start to ride it, and continue riding until he is h km past the other, when he, in turn, will leave the bike and continue again on foot. This process will continue until they reach Brian’s house. They set out together with Adam riding and Brian walking. Both of them walk at a uniform speed of 4 km/hr and ride at a uniform speed of 16 km/hr. What is the minimum time for the journey, as measured by the last one to reach the destination? If the bike changes hands twice during the journey what is the distance h?
(Problem prompted by the Leapfrog chapter in the book Number Crunching by Paul J. Nahin, Princeton University Press, 2011).
Frequency of numbers in Pascal's triangle
In Pascal’s triangle the number 1 appears infinitely many times. All other numbers will appear a finite amount of times*.
The number 2 appears just once.
The number 3 appears twice.
What is the first number that appears exactly 3 times?
What is the first number that appears exactly 4 times?
Can you find a number that appears exactly 5 times?
Can you find a number that appears exactly 6 times?
Can you find a number that appears more than 6 times?
*Can you prove this?
Circle Square
The diagram shows four circles of radius 1/4 in a unit square
A circle can fit in the gap in the middle.
What is the radius of the circle in the middle? What proportion of the original square is covered by the circle in the middle?
If you placed eight spheres of radius 1/4 in a unit cube a sphere can fit in the middle.What is the radius of the sphere in the middle? What proportion of the original cube is covered by the sphere in the middle?
What happens in 4 dimensions?
And in higher dimensions how big does the diameter of the hypersphere in the gap get and what proportion of the hypercube is covered?
Not all triangles are perfect, but...
The first two perfect numbers are 6 and 28
A number is perfect if it is equal to the sum of its factors other than itself: e.g. 6 = 1 + 2 + 3 and 28 = 1+ 2 + 4 + 7 + 14.
The first four triangular numbers are 1, 3, 6, 10. Both 6 and 28 are triangular numbers. Are all perfect numbers triangular?
The first four hexagonal numbers are 1, 6, 15, 28. Both 6 and 28 are hexagonal numbers. Are all perfect numbers hexagonal?
The IMA would like to thank Mathematics in Education and Industry (MEI) for supplying the three puzzles above. If you enjoyed these, more puzzles can be found on the MEI website by following the links below.
MEI Maths Item of the Month Archive 2010
MEI Maths Item of the Month Archive 2009
MEI Maths Item of the Month Archive 2008
MEI Maths Item of the Month Archive 2007
MEI Maths Item of the Month Archive 2006
Bee-line 1
A freight train 200m long travels along a straight track at constant speed. In the time it takes the train to move a distance equal to its own length, a bee, initially sitting at the rear of the train, flies forward to the front of the train, turns instantly and returns to the rear, flying at constant speed all the while. What is the total distance travelled by the bee?
Bee-line 2
A bee is sitting at a distance, d, from a man. At time t=0 the bee pursues the man at a constant acceleration of 1. The man runs in a straight line away from the bee with an acceleration of 0.3t. What is the largest initial separation, d, for which the man can’t quite escape the bee?
Boxes
Play against the computer by clicking on two dots to create a single vertical or horizontal line.
The player who draws the fourth and final line that completes a box wins the box and draws another line. You always draw another line after you complete a box so it’s possible to draw several lines during one turn.The game continues until every possible box has been made and there are no more dots to be joined.
To win you must have completed the most boxes.
Chicken Nuggets
At a fast-food take-away you can order chicken nuggets in boxes containing six, nine or twenty nuggets. What is the largest number of nuggets you cannot buy with these boxes?
Circular Billiards
Imagine a circular billiard table of unit radius, with centre at (x,y ) coordinates (0,0). It has two billiard balls on it. Ball A is at coordinates (-0.8, -0.1); ball B at (0.6, 0.4). What are the coordinates of the point(s) on the cushion to which one must hit ball A so that it bounces off the cushion once and then hits ball B? Explore the possibilities.
Peg
Try to clear the board by jumping pegs over other pegs. The "jumped" peg will be removed from the field. If there's more then one move available, just choose which move you would like to do. You win by getting it down to just one peg (The best solution is with this peg in the centre).
Radioactive Dice
Question: What is the half-life of radioactive dice?
No, really! It’s a serious question (well, half-serious, anyway!)
Definitions: By definition, a radioactive die is one that spontaneously disintegrates if it’s thrown six uppermost. And the half-life is the length of time you have to keep throwing lots of radioactive dice until you are left with exactly half the number you started with (ok, I know you don’t actually have any to start with – use your imagination!). “Time” here is measured in number of throws, of course.
Snow Plough
One day it started snowing steadily. A snow-plough started out at noon, going two miles in the first hour and one mile in the second hour. What time did it start snowing?
(From: Agnew, Ralph P. Differential Equations. McGraw-Hill. 1960)
The Sultan's Daughters
A sultan has 100 daughters. A commoner may be given a chance to marry one of the daughters, but he must first pass a test. He will be presented with the daughters one at a time. As each one comes before him she will tell him the size of her dowry, and he must then decide whether to accept or reject her (he is not allowed to return to a previously rejected daughter). However, the sultan will only allow the marriage to take place if the commoner chooses the daughter with the highest dowry. If he gets it wrong he will be executed! The commoner knows nothing about the distribution of dowries. What strategy should he adopt?
Please send your solutions and comments to webmaster@ima.org.uk.
